Open AccessON AN INTEGRAL INEQUALITY OF R. BELLMAN
Author(s) -
Horst Alzer
Publication year - 1991
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.22.1991.4597
Subject(s) - mathematics , combinatorics , int , inequality , discrete mathematics , mathematical analysis , computer science , operating system
We prove: if $u$ and $v$ are non-negative, concave functions defined on $[0, 1]$ satisfying
\[\int_0^1 (u(x))^{2p} dx =\int_0^1 (v(x))^{2q} dx=1, \quad p>0, \quad q>0,\]
then
\[\int_0^1(u(x))^p (v(x))^q dx\ge\frac{2\sqrt{(2p+1)(2q+1)}}{(p+1)(q+1)}-1.\]